We next measure the radial distribution function (RDF), \(g(r)\) , the probability of finding a pair of individuals at a separation distance \(r\) , relative to random movements. \(g(r) = 1\)  indicates no pair correlations, where \(g(r)>1\)  and \(g(r)<1\)  correspond to attractive and repulsive pair interactions respectively. Figure 1c plots RDF for all datasets, finding similar patterns across different datasets. First, \(g(r)\)  is consistently peaked around \(r_0\approx 0.8m\) , suggesting a typical social contact distance regardless of experimental settings. Second, individuals are unlikely to appear at very short distances around \(r < 0.3m\) , due to physical constraints of body size. Third, social interaction is persistent over a broad range of distances from \(r = 0.3m\)  through \(2-3\)\(m\) . On the other hand, the peak values of \(g(r)\)  vary significantly for D1A-D1B, indicating variation in social interaction strength. Not only are interactions weakest in D2, but its RDF also converges to unity at long distances, a typical pattern for gas. In contrast, we find \(g(r)\)  is lower than 1 asymptotically for D1 datasets, a phenomenon often observed in the liquid-vapor coexistence phase [1]}, [2]}, [3]}. This finding implies that D1 has a different macroscopic phase compared to the dataset D2, evidenced by social clustering due to stronger interaction [4]}, [5]}.
<FIGURE>